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Accurate Many-Body Repulsive Potentials for Density-Functional Tight Binding from Deep Tensor Neural Networks

Martin Stöhr, Leonardo Rafael Medrano Sandonas, Alexandre Tkatchenko

Published Jul 30, 2020• 94 citations
Publisher

Abstract

We combine density-functional tight binding (DFTB) with deep tensor neural networks (DTNN) to maximize the strengths of both approaches in predicting structural, energetic, and vibrational molecular properties. The DTNN is used to construct a nonlinear model for the localized many-body interatomic repulsive energy, which so far has been treated in an atom-pairwise manner in DFTB. Substantially improving upon standard DFTB and DTNN, the resulting DFTB-NNrep model yields accurate predictions of atomization and isomerization energies, equilibrium geometries, vibrational frequencies, and dihedral rotation profiles for a large variety of organic molecules compared to the hybrid DFT-PBE0 functional. Our results highlight the potential of combining semiempirical electronic-structure methods with physically motivated machine learning approaches for predicting localized many-body interactions. We conclude by discussing future advancements of the DFTB-NNrep approach that could enable chemically accurate electronic-structure calculations for systems with tens of thousands of atoms.

DOI
10.1021/acs.jpclett.0c01307

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The introduction of machine-learning (ML) algorithms to quantum mechanics enables rapid evaluation of otherwise intractable expressions at the cost of prior training on appropriate benchmarks. Many computational bottlenecks in the evaluation of accurate electronic structure theory could potentially benefit from the application of such models, from reducing the complexity of the underlying wave function parameter space to circumventing the complications of solving the electronic Schrödinger equation entirely. Applications of ML to electronic structure have thus far been focused on learning molecular properties (mainly the energy) from geometric representations. While this line of study has been quite successful, highly accurate models typically require a “big data” approach with thousands of training data points. Herein, we propose a general, systematically improvable scheme for wave function-based ML of arbitrary molecular properties, inspired by the underlying equations that govern the canonical approach to computing the properties. To this end, we combine the established ML machinery of the t-amplitude tensor representation with a new reduced density matrix representation. The resulting model provides quantitative accuracy in both the electronic energy and dipoles of small molecules using only a few dozen training points per system.
We present a machine learning (ML) framework for predicting Green’s functions of molecular systems, from which photoemission spectra and quasiparticle energies at quantum many-body level can be obtained. Kernel ridge regression is adopted to predict self-energy matrix elements on compact imaginary frequency grids from static and dynamical mean-field electronic features, which gives direct access to real-frequency many-body Green’s functions through analytic continuation and Dyson’s equation. Feature and self-energy matrices are represented in a symmetry-adapted intrinsic atomic orbital plus projected atomic orbital basis to enforce rotational invariance. We demonstrate good transferability and high data efficiency of the proposed ML method across molecular sizes and chemical species by showing accurate predictions of density of states (DOS) and quasiparticle energies at the level of many-body perturbation theory ( GW ) or full configuration interaction. For the ML model trained on 48 out of 1995 molecules randomly sampled from the QM7 and QM9 data sets, we report the mean absolute errors of ML-predicted highest occupied and lowest unoccupied molecular orbital energies to be 0.13 and 0.10 eV, respectively, compared to GW @PBE0. We further showcase the capability of this method by applying the same ML model to predict DOS for significantly larger organic molecules with up to 44 heavy atoms.
The introduction of machine-learning (ML) algorithms to quantum mechanics enables rapid evaluation of otherwise intractable expressions at the cost of prior training on appropriate benchmarks. Many computational bottlenecks in the evaluation of accurate electronic structure theory could potentially benefit from the application of such models, from reducing the complexity of the underlying wave function parameter space to circumventing the complications of solving the electronic Schrödinger equation entirely. Applications of ML to electronic structure have thus far been focused on learning molecular properties (mainly the energy) from geometric representations. While this line of study has been quite successful, highly accurate models typically require a “big data” approach with thousands of train- ing data points. Herein, we propose a general, systematically improvable scheme for wave function-based ML of arbitrary molecular properties, inspired by the underlying equations that govern the canonical approach to computing the properties. To this end, we combine the established ML machinery of the t-amplitude tensor representation with a new reduced density matrix representation. The resulting model provides quantitative accuracy in both the electronic energy and dipoles of small molecules using only a few dozen training points per system.
Density functional theory (DFT) is a cornerstone of computational chemistry and materials science, but its computational cost limits its use in large-scale and high-throughput applications. While machine learning has accelerated energy prediction for specific molecular classes, transferable prediction of electron density across diverse chemical spaces remains challenging. Here, we present a universal framework based on Fourier Neural Operators (FNOs) that directly learns the mapping from external potentials to electron density distributions. Unlike conventional approaches that rely on explicit atomic orbitals, basis sets, or handcrafted descriptors, the proposed method captures global electronic interactions and long-range correlations through operator learning in the spatial-frequency domain. Trained on datasets spanning multiple elements and molecular geometries, the model achieves zero-shot generalization to entirely unseen molecular systems and accurately predicts their electron densities without retraining. This transferability arises from the intrinsic ability of FNOs to represent global structure in continuous fields. Our work establishes neural operator learning as a promising route for fast, accurate, and transferable electronic structure prediction, with potential applications in high-throughput screening and chemical space exploration.
In this work we propose an artificial neural network functional to the ground-state energy of fermionic interacting particles in homogeneous chains described by the Hubbard model. Our neural network functional was proven to have an excellent performance: it deviates from numerically exact calculations by less than 0.15% for a vast regime of interactions and for all the regimes of filling factors and magnetizations. When compared to analytical functionals, the neural functional was found to be more precise for all the regimes of parameters, being particularly superior at the weakly interacting regime: where the analytical parametrization fails the most, ~7%, against only ~0.1% for the neural network. We have also applied our homogeneous functional to finite, localized impurities and harmonically confined systems within density-functional theory (DFT) methods. The results show that while our artificial neural network approach is substantially more accurate than other equivalently simple and fast DFT treatments, it has similar performance than more costly DFT calculations and other independent many-body calculations, at a fraction of the computational cost.
Computing dynamical distributions in quantum many-body systems represents one of the paradigmatic open problems in theoretical condensed matter physics. Despite the existence of different techniques both in real-time and frequency space, computational limitations often dramatically constrain the physical regimes in which quantum many-body dynamics can be efficiently solved. Here we show that the combination of machine-learning methods and complementary many-body tensor network techniques substantially decreases the computational cost of quantum many-body dynamics. We demonstrate that combining kernel polynomial techniques and real-time evolution, together with deep neural networks, allows to compute dynamical quantities faithfully. Focusing on many-body dynamical distributions, we show that this hybrid neural-network many-body algorithm, trained with single-particle data only, can efficiently extrapolate dynamics for many-body systems without prior knowledge. Importantly, this algorithm is shown to be substantially resilient to numerical noise, a feature of major importance when using this algorithm together with noisy many-body methods. Ultimately, our results provide a starting point towards neural-network powered algorithms to support a variety of quantum many-body dynamical methods, that could potentially solve computationally expensive many-body systems in a more efficient manner.